scholarly journals Pseudocomplements in groupoids

1978 ◽  
Vol 26 (2) ◽  
pp. 209-219 ◽  
Author(s):  
K. Nirmala Kumari Amma
Keyword(s):  
Boolean Algebra ◽  
Prime Ideals ◽  
Minimal Prime

AbstractThis paper is devoted to a study of pseudocomplements in groupoids. A characterization of an intraregular groupoid is obtained in terms of prime ideals. It is proved that the set of dense elements of an intraregular groupoid S with 0 is the intersection of all the maximal filters of S and that the set of normal elements of an intraregular groupoid closed for pseudocomplements forms a Boolean algebra under natural operations. It is shown that the pseudocomplement of an ideal of an intraregular groupoid with 0 is the intersection of all the minimal prime ideas not containing it.

scholarly journals A characterization of minimal prime ideals

1998 ◽  
Vol 40 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Gary F. Birkenmeier ◽  
Jin Yong Kim ◽  
Jae Keol Park
Keyword(s):  
Positive Integer ◽  
Prime Ideal ◽  
Commutative Ring ◽  
Minimal Prime Ideal ◽  
Prime Ideals ◽  
Sheaf Representations ◽  
Minimal Prime

AbstractLet P be a prime ideal of a ring R, O(P) = {a ∊ R | aRs = 0, for some s ∊ R/P} | and Ō(P) = {x ∊ R | xn ∊ O(P), for some positive integer n}. Several authors have obtained sheaf representations of rings whose stalks are of the form R/O(P). Also in a commutative ring a minimal prime ideal has been characterized as a prime ideal P such that P= Ō(P). In this paper we derive various conditions which ensure that a prime ideal P = Ō(P). The property that P = Ō(P) is then used to obtain conditions which determine when R/O(P) has a unique minimal prime ideal. Various generalizations of O(P) and Ō(P) are considered. Examples are provided to illustrate and delimit our results.

scholarly journals A characterization of semi-prime ideals in near-rings

1983 ◽  
Vol 35 (2) ◽  
pp. 194-196 ◽  
Author(s):  
N. J. Groenewald
Keyword(s):  
Prime Ideal ◽  
Prime Radical ◽  
Prime Ideals ◽  
Minimal Prime

AbstractIt is well-known that in any near-ring, any intersection of prime ideals is a semi-prime ideal. The aim of this note is to prove that any ideal is a prime ideal if and only if it is equal to its prime radical. As a consequence of this we have any semi-prime ideal I in a near-ring N is the intersection of minimal prime ideals of I in N and that I is the intersection of all prime ideals containing I.

scholarly journals A Characterization of semiprime ideals in near-rings

1982 ◽  
Vol 32 (2) ◽  
pp. 212-214 ◽  
Author(s):  
V. Sambasiva Rao
Keyword(s):  
Prime Ideals ◽  
Semiprime Ideal ◽  
Minimal Prime

AbstractIt is well known that in any near-ring, any intersection of prime ideals is a semiprime ideal. The aim of this paper is to prove that any semiprime ideal I in a near-ring N is the intersection of all minimal prime ideals of I in N. As a consequence of this we have any seimprime ideal I is the intersectionof all prime ideals containing I.

The Minimal Prime Spectrum of a Commutative Ring

1971 ◽  
Vol 23 (5) ◽  
pp. 749-758 ◽  
Author(s):  
M. Hochster
Keyword(s):  
Topological Space ◽  
Commutative Ring ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Prime Spectrum ◽  
Topological Characterization ◽  
Completely Regular ◽  
Minimal Prime

We call a topological space X minspectral if it is homeomorphic to the space of minimal prime ideals of a commutative ring A in the usual (hull-kernel or Zariski) topology (see [2, p. 111]). Note that if A has an identity, is a subspace of Spec A (as defined in [1, p. 124]). It is well known that a minspectral space is Hausdorff and has a clopen basis (and hence is completely regular). We give here a topological characterization of the minspectral spaces, and we show that all minspectral spaces can actually be obtained from rings with identity and that open (but not closed) subspaces of minspectral spaces are minspectral (Theorem 1, Proposition 5).

scholarly journals Congruence pairs of principal MS-algebras and perfect extensions

2020 ◽  
Vol 70 (6) ◽  
pp. 1275-1288
Author(s):  
Abd El-Mohsen Badawy ◽  
Miroslav Haviar ◽  
Miroslav Ploščica
Keyword(s):  
Boolean Algebra ◽  
Congruence Lattices ◽  
Descriptive Solution ◽  
The One ◽  
Special Case ◽  
Congruence Pair

AbstractThe notion of a congruence pair for principal MS-algebras, simpler than the one given by Beazer for K2-algebras [6], is introduced. It is proved that the congruences of the principal MS-algebras L correspond to the MS-congruence pairs on simpler substructures L°° and D(L) of L that were associated to L in [4].An analogy of a well-known Grätzer’s problem [11: Problem 57] formulated for distributive p-algebras, which asks for a characterization of the congruence lattices in terms of the congruence pairs, is presented here for the principal MS-algebras (Problem 1). Unlike a recent solution to such a problem for the principal p-algebras in [2], it is demonstrated here on the class of principal MS-algebras, that a possible solution to the problem, though not very descriptive, can be simple and elegant.As a step to a more descriptive solution of Problem 1, a special case is then considered when a principal MS-algebra L is a perfect extension of its greatest Stone subalgebra LS. It is shown that this is exactly when de Morgan subalgebra L°° of L is a perfect extension of the Boolean algebra B(L). Two examples illustrating when this special case happens and when it does not are presented.

On minimal prime ideals of commutative Bezout rings

1999 ◽  
Vol 51 (7) ◽  
pp. 1129-1134
Author(s):  
B. V. Zabavskii ◽  
A. I. Gatalevich
Keyword(s):  
Prime Ideals ◽  
Minimal Prime

scholarly journals Characterizations of m-Normal Nearlattices in terms of Principal n-Ideals

2013 ◽  
Vol 38 ◽  
pp. 49-59
Author(s):  
MS Raihan
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Single Element ◽  
Finitely Generated ◽  
Fixed Element ◽  
Minimal Prime

A convex subnearlattice of a nearlattice S containing a fixed element n?S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal n-ideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In this paper, we include several characterizations of those Pn(S) which form m-normal nearlattices. We also show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,…., Pm of S, Po ? … ? Pm = S. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16548 Rajshahi University J. of Sci. 38, 49-59 (2010)

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